I'd say the main difference, aside from the fact that "animar" has a broader meaning than "alentar", is the register. You wouldn't say "alentar" in a casual register or without a poetic meaning.
I also use a lot "animar" in a sense similar to "cheer up." For example, when someone is feeling down or depressed, you say "¡Venga, anímate!" which you can translate as "C'mon, cheer up!". As an example where "Alentar" could fit: in a report about people cheering for a football team, the reporter might say "Una multitud de personas alentaron a la selección española de fútbol en su debút." which can be translated as "A crowd of people cheered the spanish football team in its debut." In this context "alentar" may be translated as the Spanish idiom "dar aliento" which literally means "to give breath".
Yesterday while reading a medical report I realized that probably, the most common use of "alentar" is the adjetive "alentador": to express that the results of a forecast or a survey are encouraging, it's not unusual to hear the word on a politician speech or from a doctor. In fact, there isn't an equivalent to "alentador" with the verb "animar" (You might hear "Los resultados de su analítica son alentadores" but you definitely won't hear that the results were "animadores")